On the distance spectral radius, fractional matching and factors of graphs with given minimum degree

Abstract: Let $D(G)=(d_{ij}){n\times n}$ be the distance matrix of a graph $G$ of order $n$. The largest eigenvalue of $D(G)$, denoted by $\mu(G)$, is called the distance spectral radius of $G$. A fractional matching of $G$ is a function $f: E(G)\to [0,1]$ such that for any $v_i\in V(G)$, $\sum{e\in E_G(v_i)}f(e)\le 1$, where $E_G(v_i)={e:e\in E(G) \ \textrm{and}\ e \ \textrm{is incident with} \ v_i}$. Let $\alpha_f(G)$ denote the fractional matching number of $G$ which is defined as $\alpha_f(G)=\max{\sum_{e\in E(G)}f(e): f\ \textrm{is a fractional matching of} \ G}$. In this paper, we establish a sharp upper bound for the distance spectral radius to guarantee $\alpha_f(G)>\frac{n-k}{2}$ in graph $G$ with given minimum degree $\delta$. Let ${G_1,G_2,G_3,\cdots}$ be a set of graphs, an ${G_1,G_2,G_3,\cdots}$-factor of a graph $G$ is a spanning subgraph of $G$ such that each component of which is isomorphic to one of ${G_1,G_2,G_3,\cdots}$. In this paper, we give a sharp upper bound on distance spectral radius of graph $G$ with given minimum degree $\delta$ to ensure that $G$ has a ${K_2, C_k}$-factor, where $k\ge3$ is an integer. In addition, we also obtain a sharp upper bound on distance spectral radius for the existence of a ${K_{1,1},K_{1,2},\cdots,K_{1,k}}$-factor with $k\ge2$ in a graph $G$ with given minimum degree $\delta$.